The affine rank minimization (ARM) problem is well known for both its applications and the fact that it is NP-hard. One of the most successful approaches, yet arguably underrepresented, is iteratively reweighted least squares (IRLS), more specifically $\mathrm{IRLS}$-$0$. Despite comprehensive empirical evidence that it overall outperforms nuclear norm minimization and related methods, it is still not understood to a satisfying degree. In particular, the significance of a slow decrease of the therein appearing regularization parameter denoted $\gamma$ poses interesting questions. While commonly equated to matrix recovery, we here consider the ARM independently. We investigate the particular structure and global convergence property behind the asymptotic minimization of the log-det objective function on which $\mathrm{IRLS}$-$0$ is based. We expand on local convergence theorems, now with an emphasis on the decline of $\gamma$, and provide representative examples as well as counterexamples such as a diverging $\mathrm{IRLS}$-$0$ sequence that clarify theoretical limits. We present a data sparse, alternating realization $\mathrm{AIRLS}$-$p$ (related to prior work under the name $\mathrm{SALSA}$) that, along with the rest of this work, serves as basis and introduction to the more general tensor setting. In conclusion, numerical sensitivity experiments are carried out that reconfirm the success of $\mathrm{IRLS}$-$0$ and demonstrate that in surprisingly many cases, a slower decay of $\gamma$ will yet lead to a solution of the ARM problem, up to the point that the exact theoretical phase transition for generic recoverability can be observed. Likewise, this suggests that non-convexity is less substantial and problematic for the log-det approach than it might initially appear.

翻译：最小化(ARM)问题在应用中是众所周知的。 最成功的方法之一 — — 虽然代表率可能很低 — — 最明显 — — 美元。 最成功的方法之一 — — 最明显的是反复重标最小方块(IRLS ), 更具体地说是 $\ mathrm{IRLS} $ - 0美元。 尽管全面的经验证据表明,它总体上超过了核规范最小化和相关方法,但它仍然不能令人满意地理解。 特别是, 此处显示的正规化参数降幅($- gamma$ 表示的敏感度) 提出了有趣的问题。 虽然通常等同于矩阵恢复,但我们在这里独立考虑ARM 的过渡。 我们调查了在对日记目标函数最小化后的特定结构和全球趋同属性($- maylor_ IR) 的最小值最小化值 。 我们扩大本地的趋同值,现在强调美元下降, 并且提供有代表性的示例,比如变差 美元 IMLS 和 美元 的递增 。